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A099308 Numbers n whose k-th arithmetic derivative is zero for some k. Complement of A099309. 16
0, 1, 2, 3, 5, 6, 7, 9, 10, 11, 13, 14, 17, 18, 19, 21, 22, 23, 25, 29, 30, 31, 33, 34, 37, 38, 41, 42, 43, 46, 47, 49, 53, 57, 58, 59, 61, 62, 65, 66, 67, 70, 71, 73, 77, 78, 79, 82, 83, 85, 89, 93, 94, 97, 98, 101, 103, 105, 107, 109, 113, 114, 118, 121, 126, 127, 129, 130 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

The first derivative of 0 and 1 is 0. The second derivative of a prime number is 0.

REFERENCES

See A003415

LINKS

T. D. Noe, Table of n, a(n) for n = 1..10000

EXAMPLE

18 is on this list because the first through fifth derivatives are 21, 10, 7, 1, 0

MATHEMATICA

dn[0]=0; dn[1]=0; dn[n_]:=Module[{f=Transpose[FactorInteger[n]]}, If[PrimeQ[n], 1, Plus@@(n*f[[2]]/f[[1]])]]; d1=Table[dn[n], {n, 40000}]; nLim=200; lst={1}; i=1; While[i<=Length[lst], currN=lst[[i]]; pre=Intersection[Flatten[Position[d1, currN]], Range[nLim]]; pre=Complement[pre, lst]; lst=Join[lst, pre]; i++ ]; Union[lst]

CROSSREFS

Cf. A003415 (arithmetic derivative of n), A099307 (least k such that the k-th arithmetic derivative of n is zero), A099309 (numbers whose k-th arithmetic derivative is nonzero for all k).

Sequence in context: A305847 A248565 A065896 * A074235 A325366 A192189

Adjacent sequences:  A099305 A099306 A099307 * A099309 A099310 A099311

KEYWORD

nonn

AUTHOR

T. D. Noe, Oct 12 2004

STATUS

approved

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Last modified January 26 05:37 EST 2020. Contains 331273 sequences. (Running on oeis4.)